It is widely known that conditional covariances of asset returns change over time. Researchers adopt many strategies to accommodate conditional heteroskedasticity. Among the most popular: (a) chopping the data into short blacks of time and assuming homoskedasticity within the blocks, (b) performing one-sided rolling regressions, in which only data from, say, the preceding five year period is used to estimate the conditional covariance of returns at a given date, and (c) two-sided rolling regressions which use, say, five years of leads and five years of lags. GARCH amounts to a one-sided rolling regression with exponentially declining weights. We derive asymptotically optimal window lengths for standard rolling regressions and optimal weights for weighted rolling regressions. An empirical model of the S&P 500 stock index provides and example.
We examine in this paper a new natural restriction on utility functions, namely that adding an unfair background risk to wealth makes risk-averse individuals behave in a more risk-averse way with respect to any other independent risk. This concept is called risk vulnerability. It is equivalent to the condition that an undesirable risk can never be made desirable by the presence of an independent, unfair risk. Moreover, under risk vulnerability, adding an unfair background risk reduces the demand for risky assets. Risk vulnerability generalizes the concept of properness (individually undesirable, independent risks are always jointly undesirable) introduced by Pratt and Zeckhauser (1987). It implies that the two first derivatives of the utility function are concave transformations of the original utility function. Under decreasing absolute risk aversion, a sufficient condition for risk vulnerability is local properness, i.e. r'' ≥ r'r, where r is the Arrow-Pratt coefficient of absolute risk aversion.
ECONOMIC DECISION MAKING UNDER UNCERTAINTY often takes place in the presence of multiple and in markets that are less than complete. As a consequence, choices about endogenous sometimes must be made while simultaneously facing one or more immutable exogenous risks that are not under the control of the agent, and that are independent of endogenous risks. It is somehow natural to assume that an exogenous deterioration in background wealth will cause an individual to take more care elsewhere. If we define a deterioration, for example, as making the individual poorer by removing a fixed amount of initial wealth, we know from Pratt (1964) that decreasing absolute risk aversion (DARA) of an individual's von Neumann-Morgenstern utility function yields this natural result. If, on the other hand, background wealth becomes riskier due to the addition of a zero-mean risk, that is also statistically independent of all other risks, behavior will be more risk averse if and only if preferences are risk vulnerable as defined by Gollier and Pratt (1996). Risk vulnerability (described below in Section 4) is a stronger notion than DARA and includes proper risk aversion (Pratt and Zeckhauser (1987)) and standard risk aversion (Eeckhoudt and Kimball (1992), Kimball (1993)) as particular cases. But a deterioration in background wealth may encompass more complicated distribution changes than the introduction of another statistically independent risk. In this paper, we examine background wealth deteriorations that take the form of both general firstand second-degree stochastic dominance changes in risk (FSD and SSD respectively). In particular, we determine conditions that are both necessary and sufficient for each of these two types of background risk changes to imply more risk-averse behavior on the part of the individual. For the case of FSD changes, this condition turns out to be Ross' stronger characterization of decreasing absolute risk aversion. In the case of general SSD changes in the distribution of background wealth, the condition derived is a stronger version (in Ross' sense) of the conditions characterizing preferences that are locally risk vulnerable in the sense of Gollier and Pratt. The necessary and sufficient conditions derived are fairly restrictive upon preferences. However, if we take as positive behavior that individuals act in a more risk-averse manner whenever the distribution of background wealth deteriorates, these conditions place canonical limits upon appropriate utility representations. At the very least, they
This paper derives the asymptotic power envelope for tests of a unit autoregressive root for various trend specifications and stationary Gaussian autoregressive disturbances. A family of tests is proposed, members of which are asymptotically similar under a general 1(1) null (allowing nonnormality and general dependence) and which achieve the Gaussian power envelope. One of these tests, which is asymptotically point optimal at a power of 50%, is found (numerically) to be approximately uniformly most powerful (UMP) in the case of a constant deterministic term, and approximately uniformly most powerful invariant (UMPI) in the case of a linear trend, although strictly no UMP or UMPI test exists. We also examine a modification, suggested by the expression for the power envelope, of the Dickey-Fuller (1979) t-statistic; this test is also found to be approximately UMP (constant deterministic term case) and UMPI (time trend case). The power improvement of both new tests is large: in the demeaned case, the Pitman efficiency of the proposed tests relative to the standard Dickey-Fuller t-test is 1.9 at a power of 50%. A Monte Carlo experiment indicates that both proposed tests, particularly the modified Dickey-Fuller t-test, exhibit good power and small size distortions in finite samples with dependent errors.
Our general subject is model determination methods and their use in the prediction of economic time series. The methods suggested are Bayesian in spirit but they can be justified by classical as well as Bayesian arguments. The main part of the paper is concerned with model determination, forecast evaluation, and the construction of evolving sequences of models that can adapt in dimension and form (including the way in which any nonstationarity in the data is modelled) as new characteristics in the data become evident. The paper continues some recent work on Bayesian asymptotics by the author and Werner Ploberger (1995), develops embedding techniques for vector martingales that justify the role of a class of exponential densities in model selection and forecast evaluation, and implements the modelling ideas in a multivariate regression framework that includes Bayesian vector autoregressions (BVAR's) and reduced rank regressions (RRR's). It is shown how the theory in the paper can be used: (i) to construct optimized BVAR's with data-determined hyperparameters; (ii) to compare models such as BVAR's, optimized BVAR's, and RRR's; (iii) to perform joint order selection of cointegrating rank, lag length, and trend degree in a VAR; and (iv) to discard data that may be irrelevant and thereby reset the initial conditions of a model.
This paper explores a one-agent Bayesian model of learning by doing and technological choice.To produce output, the agent can choose among various technologies.The beneficial effects of learning by doing are bounded on each technology, and so long-run growth in output can take place only if the agent repeatedly switches to better technologies.As the agent repeatedly uses a technology, he learns about its unknown parameters, and this accumulated expertise is a form of human capital.But when the agent switches technologies, part of this human capital is lost.It is this loss of human capital that may prevent the agent from moving up the quality ladder of technologies as quickly as he can, since the loss is greater the bigger is the technological leap.We analyze the global dynamics.We find that a human-capital-rich agent may find it optimal to avoid any switching of technologies, and therefore to experience no long-run growth.On the other hand, a human-capital-poor agent, who because of his lack of skill is not so attached to any particular technology, can find it optimal to switch technologies repeatedly, and therefore enjoy long-run growth in output.Thus the model can give rise to overtaking.
We consider the situation where a single consumer buys a stream of goods from different sellers over time. The true value of each seller's product to the buyer is initially unknown. Additional information can be gained only by experimentation. For exogeneously given prices the buyer's problem is a multi-armed bandit problem. The innovation in this paper is to endogenize the cost of experimentation to the consumer by allowing for price competition between the sellers. The role of prices is then to allocate intertemporally the costs and benefits of learning between buyers and sellers. We examine how strategic aspects of the oligopoly model interact with the learning process. All Markov perfect equilibria (MPE) are efficient. We identify an equilibrium which besides its unique robustness properties has a strikingly simple, seemingly myopic pricing rule. Prices below marginal cost emerge naturally to sustain experimentation. Intertemporal exchange of the gains of learning is necessary to support efficient experimentation. We analyze the asymptotic behavior of the equilibria.
mutation.) Kandori, Mailath, and Rob (henceforth KMR) first provide a useful general theorem concerning the stationary distribution of strategies under Darwinian dynamics. They then divide the analysis of the 2 x 2 game into three cases: dominant strategy games (e.g., prisoners' dilemma), coordination games, and games with no symmetric pure strategy equilibrium (e.g., battle of the sexes). We refer to these as DS, C, and NP games. In each case, KMR claim that, as the rate of mutation vanishes, the stationary distribution of strategies converges to a symmetric Nash equilibrium. They emphasize C games, which have two symmetric Nash equilibria, and characterize the conditions under which the distribution converges to the risk dominant equilibrium. In this note, we argue that while their formal conclusions for C games are correct, their results for DS and NP games are valid only for large populations of players. In small populations, Darwinian dynamics may produce non-Nash outcomes in these two cases. Section 1 summarizes the KMR model, and Section 2 provides examples of 2 x 2 games in which Darwinian dynamics generate non-Nash outcomes. A theorem in Section 3 describes the Darwinian equilibrium of any 2 x 2 game.
ABSTRACT The use of predetermined variables to represent public information and time‐variation has produced new insights about asset pricing models, but the literature on mutual fund performance has not exploited these insights. This paper advocates conditional performance evaluation in which the relevant expectations are conditioned on public information variables. We modify several classical performance measures to this end and find that the predetermined variables are both statistically and economically significant. Conditioning on public information controls for biases in traditional market timing models and makes the average performance of the mutual funds in our sample look better.
ABSTRACT An analysis of new buy and sell recommendations of stocks by security analysts at major U.S. brokerage firms shows significant, systematic discrepancies between prerecommendation prices and eventual values. The initial return at the time of the recommendations is large, even though few recommendations coincide with new public news or provide previously unavailable facts. However, these initial price reactions are incomplete. For buy recommendations, the mean postevent drift is modest (+2.4%) and short‐lived, but for sell recommendations, the drift is larger (−9.1%) and extends for six months. Analysts appear to have market timing and stock picking abilities.